Calculate the required probabilities for the normal
distributions with the parameters specified in parts a through
e.
a. μ = 5, σ = 4; calculate P(0 < x < 6).
P(0 < x < 6) = ______ . (Round to four decimal places as needed.)
b. μ = 5, σ = 5; calculate P(0 < x < 6).
P(0 < x < 6) = ______ . (Round to four decimal places as needed.)
c. μ = 4, σ = 4; calculate P(0 < x < 6).
P(0 < x < 6) = ______ . (Round to four decimal places as needed.)
d. μ = 6, σ = 2; calculate P(x > 1).
P(x > 1) = ______ . (Round to four decimal places as needed.)
e. μ = 0, σ = 2; calculate P(x > 1).
P(x > 1) = ______ . (Round to four decimal places as needed.)
Solution :
a.
P(0 < x < 6) = P[(0 - 5)/ 4) < (x -
) /
<
(6 - 5) / 4) ]
= P(-1.25 < z < 0.25)
= P(z < 0.25) - P(z < -1.25)
= 0.5987 - 0.1056
= 0.4931
P(0 < x < 6) = 0.4931
b.
P(0 < x < 6) = P[(0 - 5)/ 5) < (x -
) /
<
(6 - 5) / 5) ]
= P(-1 < z < 0.2)
= P(z < 0.2) - P(z < -1)
= 0.5793 - 0.1587
= 0.4206
P(0 < x < 6) = 0.4206
c.
P(0 < x < 6) = P[(0 - 4)/ 4) < (x -
) /
<
(6 - 4) / 4) ]
= P(-1 < z < 0.5)
= P(z < 0.5) - P(z < -1)
= 0.6915 - 0.1587
= 0.5328
P(0 < x < 6) = 0.5328
d.
P(x > 1) = 1 - P(x < 1)
= 1 - P[(x -
) /
< (1 - 6) / 2)
= 1 - P(z < -2.5)
= 1 - 0.0062
= 0.9938
P(x > 1) = 0.9938
e.
P(x > 1) = 1 - P(x < 1)
= 1 - P[(x -
) /
< (1 - 0) / 2)
= 1 - P(z < 0.5)
= 1 - 0.6915
= 0.3085
P(x > 1) = 0.3085
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